Optimal PSK Signaling over Stationary Rayleigh Fading Channels

We consider a stationary Rayleigh flat-fading channel with temporal correlation and a compactly supported power spectral density of the channel fading process. We assume that the channel state is unknown to both transmitter and receiver, while the law of the channel is presumed to be known to the receiver. Given a set of fixed signaling sequences, the optimum input distribution, with respect to the achievable rate, has the property of a constant Kullback-Leibler distance between the output distribution and a mixture of the output distributions conditioned on the different input sequences. Based on this, we determine the set of optimum input distributions for PSK signaling. In addition, we identify the special case of transmitting one pilot symbol to acquire a phase reference as being included in the set of optimum input distributions. We derive an integral expression for the capacity constrained to PSK signaling depending on the autocorrelation of the channel and the SNR. Evaluation of the asymptotic high SNR behavior shows a loss in the constrained capacity with respect to the case of perfect channel knowledge corresponding to at least one signaling dimension, i.e., the information transmitted by one symbol.